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Theorem oteqex2 4448
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
Assertion
Ref Expression
oteqex2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )

Proof of Theorem oteqex2
StepHypRef Expression
1 opeqex 4447 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  _V )  <->  ( <. R ,  S >.  e.  _V  /\  T  e.  _V )
) )
2 opex 4427 . . 3  |-  <. A ,  B >.  e.  _V
32biantrur 493 . 2  |-  ( C  e.  _V  <->  ( <. A ,  B >.  e.  _V  /\  C  e.  _V )
)
4 opex 4427 . . 3  |-  <. R ,  S >.  e.  _V
54biantrur 493 . 2  |-  ( T  e.  _V  <->  ( <. R ,  S >.  e.  _V  /\  T  e.  _V )
)
61, 3, 53bitr4g 280 1  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817
This theorem is referenced by:  oteqex  4449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823
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